Police can see your private Google searches, but only with a warrant.
If you are searching illegal things and get flagged, then police can be tipped off and get a warrant from a judge to have Google and even your ISP or internet service provider to provide the police with your private searches.
Even if you delete your own search history, Google also still keeps your search history on record and your ISP also keeps a log of websites you visited and ip addresses that were accessed with your computer or device.
Your ISP can also give police your address as well.
While there's really nothing that you're not allowed to search on Google, some search terms can get you flagged and possibly investigated.
Searching for terrorism, harassing, bomb making, threats, how to murder someone etc or how to assassinate someone etc could get you into trouble.
Although Google does not allow content that mainly facilitates the promotion or the sale of regulated services and goods, like gambling, alcohol, tobacco, pharmaceuticals, unapproved supplements, fireworks, weapons or health and medical devices.
Google also has content policies against dangerous content and deceptive practices.
But there's not actual banned keywords that you can't search, but some things should not be searched on google or other search engines.
Google searches that can get you flagged are searching for things such as bomb making, illegal downloads, terrorism, and child pornography.
Searching Google search for the words terrorism and child pornography once or a few times is unlikely to get you flagged and get you into trouble as it could be seen as you just seeking out the terms for knowledge and not for ill intent.
But if you search for these terms often then it could get you lagged.
Other Google searches that could get you flagged are terms such as child exploitation and even hacking.
Deleting your browsing history is not a crime unless you had searched with intent for illegal things and you deleted the browsing history to tamper with evidence.
If you delete browsing history after being charged with a crime or being investigated with a crime then it can be illegal to delete that browsing history and be charged as tampering with evidence.
Even when you delete your browsing history, it can often still be found through Googles records or even through forensic software.
Search history is not ever truly deleted even when you delete your search history from your computer or devices as Google and other search engines keep your search history in their systems for a certain amount of time in case they have to give it to law enforcement.
While you can delete your search history from your browser and computer or device, it still remains with Google and other search engines that you used.
The FBI can see your deleted search history through use of forensic software, which can recover deleted files in most cases.
The FBI can also issue a subpoena to your internet service provider or Google or any other search engine company you used to get the search history.
Deleting your search history only deletes the search history from your computer or device, but not from the search engines system itself.
Your ISP also has records on file that they keep for a certain time of your search history and any files you may have downloaded or websites you viewed.
The forensic software that FBI and law enforcement use can scan your devices for data that has not been overwritten and so they can recover a wide range deleted data including your search history.
Police can see Google searches if the police have a warrant or subpoena and submit that warrant or subpoena to Google to access your searches.
Or if Google detects illegal images on your Gmail account or in your Google drive or Google account through automatic systems, then Google can and often will report your account to law enforcement and they will then look into your searches and data and if they find something illegal they will show up at your door eventually with a warrant to seize your devices, search your home and possibly arrest you.
You can get into trouble for googling illegal things, although it will depend on what you're actually searching that is illegal.
Simply googling some illegal things won't always get you into trouble.
Although if you google something like Child Porn and your intent is to find and look at and save that child porn then you can get into trouble.
Google also has systems in place that can automatically detect potential child porn content even when it's sent through their Gmail service.
If google detects child porn or possible child porn being sent through your Gmail account or you save child porn or trade child porn or other illegal things on your google account including google drive account they will contact authorities.
Although simply Googling the term child pornography is not illegal in itself.
As someone searching the term child pornography may be simply trying to find out about how to report the child porn or looking on the law about it.
Also while viewing and or possessing child porn is also illegal, you won't always get into trouble if you happen to stumble upon child porn.
There has to be intent.
Also searching for tools or even guides for engaging in cybercrimes such as hacking is also illegal.
And even if your search intent is not malicious, searching for topics like how to make a homemade bomb, etc can also trigger an investigation and can place you on a watch list.
Police can also obtain a warrant to access your search history from Google or your ISP and they can provide that even if you delete your search history or destroy your device or computer.
Some keywords and phrases can also sometimes trigger automatic flags from law enforcement in some cases.